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MAT-152 7.1b: Applications of the Substitution Method

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hello everybody Welcome to chapter 7 1 Part B we're still doing systems using substitution but this chapter is going to go over the chapters is going to talk about the Apple you guys on that means word problems Emily what we're going to get from this lesson is there an understanding solutions to systems is the point where the two equations are equal to each other and that means a lot of different things problems and we can find things like just an ounce of money and we can find where two functions kind of like break even with each other where one function starts to exceed another function we can find a lot of interesting things Chris example we're going to talk about is an amount of Interest so we're going to try this one interest K & R does a total of $12,000 is invested into funds to fund paying 9 percent interest and 11% Pinterest the yearly interest $1,180 and what we're finding out is how much is invested at each rate in other words how much money did you put in each funds that's what we don't know yet however we're going to write out in words but we do know we do know that 9% all of the fun we don't know I'm going to call this send x-ray cuz you don't know the fine but don't know how much it is a plus 11% of fund why cuz they don't know what that funded we don't know how much we put need to find what's going to be the total amount invested all right now let's work with that interest rate right we know that we have 9% interest 11 percentage equals that total interest alright and here's the thing what we don't know is we don't know how much was invested at 9% and we don't know how much was invested at 11% we don't know those amount so that's what we're going to label our Acts send a text and fund why we don't know fund actually don't know fund why so we're going to say that's the total amount invested a part of me we're going to call at 12. because that's how much was invested Total Wine percent of the X fund + 11% refund was the total amount of interest that was 1180 alright and now we've got some numbers you can sort of work with but we can make it even easier to use so here's what we're going to do we're going to rerun or system a separate sheet of paper and we are going to multiply the bottom one by 100 so we can easier numbers to work with just looking to have Expo 5 equals 12000 start with 0.9 x + 0.11 y equals 1180 but to make these numbers easy to work with I don't care about you like dealing with hundreds so what I'm going to do is I'm going to multiply both sides of this equation by 100 always allowed both sides of an equation as long as you do it supposed to say if you haven't fundamentally change the equation you just made it easier to work with so my new system is going to look like this I'm going to have like for a little arrow changing it to we're going out X Plus y equals 12 Frozen and I do all this mess I get 9x + 11y wow that's a lot easier equals 100 hang on let me think about that for a second okay I hundred 18th alright awesome kind of look at this and see that we almost got a variable isolated equation take out top equation and solve the top for Action going to write that stuff in a box so we understand what happened here so the top 4 x that's what we're going to do I'm going to take the top equation x + y equals 12000 and I'm going to subtract y on both side when I do that I'm going to get 12000 - Y and oh my goodness check it out self rack and now I have a piece I have something some stuff that I can put into the bomb equation for x so I can get rid of that acts and put this stuff in and it's exactly what I'm going to do I'm going to substitute into the bottom cat litter box around that stuff and let's do it 9 times 12000 minus y + 11 y 118000 I do some distribution and I get 108000 - 9 y + 11 y equals 118000 steps at once here I'm going to subtract my 108,000 from both sides and I'm going to combine - 9y + Milan why and when I do going to get to y equals 10000 y equals 5000 what we can do is we can take this and we can substitute it into results for X cuz now we can just put y right there that 5000 right there and get axe super fast and let's do that we're going to say substitutes two remaining equation I guess I should say Trina it really doesn't matter I'm going to cut the word remaining I'm going to say equation doesn't matter can be either one whatever I just personally think that this is a sensible thing to plug into because that's going to be really easy alright stop talking what do snap 5000 + 4y it looks like and now and I saw 4 x I get 7000 so... What I'm really saying with this is that I remember my 7000 was my ex found it was my knife so what I'm saying is $7,000 was invested invested at 9 percent interest and $5,000 was invested 11% interest there we have it all right that's our first problem let's move on to the next one break even Point snacks be like fun so deposit right the problem then you can call me the problem down since video takes less time alright Sabrina. This word problem if you like a second and pause your video to write it all down please do small business invest $10,000 in equipment to produce a product each unit of the product costs $0.65 to make and is sold for $1.20 how many items must be sold in order for business to break even I hear I would like to remind you that first of all the basic profit equation on a profit comes from revenue Minus cost in what were interested in with this break even Point Break even is going to happen when prophets no longer negative when profit is zero because after profit is zero directly after that is when it's going to start because they start to make some money so the break-even point is when profit is zero so let's feelings you for-profit I'm not see it what were you can do with these if I add see on both sides I get the equation R equals c so really what the break-even point means is when Revenue Eagles Past so what I'm going to do first of all those were going to make a revenue equation and it cost equation and we're going to call them both see and then we're going to substitute in so here's what we're going to do that's right everything out in words wants more just because it's always helpful to do it's a good strategy good skill to have when you're dealing with application problems we know Revenue the amount of money coming in is derived from the amount made so like money made from each item multiply that by how many items salt and our cost includes a few different things not only are we talking about the cost to make each item cost per unit cost per item times the number of items made but we also have to take into account initial Cox business is typically don't just rice from without any money being spent on them usually have start-up costs or initial cost let's fill in all of these words are we know not the number of items are sold yet that's going to be are very lucky to be your ex but we do know how much they make from each item which is 120 and we do know that it cost him $0.65 and we do know that initially they had to spend 10000 steps that look like this what's it up as our equals 1.2 x $1 20 for the number of items I don't know cost is $0.65 0.65 times a number of items I don't know yet plus how much they spent initially which was a $10,000 investment I said got two equations great remember I'm interested in what I'm going to do with these isn't going to find out when are equals see when this function equals this function so here's what I can do substitute in 1.2 x 4 see that's exactly what we're going to do we're going to write that stuff out so then we can truly find when are equals t 1.2 x equals 0.265 x + 10,000 two step equation or pretty salad with this we should get remove my paper up here we should be getting 0.55 x equals 10000 why don't we do some division we are going to signs that X is approximately I'm going to round up here you should round up as well when you put a snare calculator 18182 units I want that too we are needing to produce 18180 units for these two functions to intersect each other and after this I'm out making 18180 to you we will start to make money this is just where we stopped losing money and then we make money after this I took break even me alright let's move on to the last example same thing you're going to see a word problem pop up feel free to posit to copy Ernie's take a second pause your video if you'd like to write down this word problem but I'll read through it this one's title of catch-up points because it's another thing systems help us find from 1990 to 2000 the population of Tennessee was increase at a rate faster than the population of Zuri models are approximately two populations are pardoning equals 44866 + 74.52 + P equals 5106 what's 43.5 T where T equals zero corresponds to 1990 when will the population of Tennessee exceed the population of Missouri according to the models from this problem matter we don't know which one of these is misery and which one of these is Tennessee that has no bearing on a right answer here the point where Tennessee is going to exceed the population of Missouri is going to be at a point where these two graphs intersect and again I cannot find that by using substitution and solving and you can see kind of an sing going on here we have a value for p and we have another value for peeing so what we can do is just take some of this stuff for p and put it in for key in the other equation eliminate that variable all together and then we'll get an equation that's going to be like two to four steps that's going to be too hard. Let's start doing that one function 4p that's our first step to draw a little box around it all right and doesn't matter how you do it but you should get an equation that looks like this either with the right side different but doesn't matter 4866 + 74.5 T equals 5106 + 43.5 t and in about three steps we can solve this if we subtract 43.5 to hear subtract 43.5 T hear subtract 4866 subtract 4866 that algebra we get that 31t equals 240 Edwin we divided by 31 we get tea is approximately rounding Street as animals 7.742 and let's keep in mind what this means we just found out we're going to do like a little... We're going to say I T equals zero sanso 1990 correct so if 1990 is when T equals zero 20 equals 7 that must be the Year 1997 and were bigger than just seven where 7:40 742 so we can stay in the year 1997 sometime in their right probably later in the year you were almost to 8, but some find a nice 97 Tennessee population will exceed Missouri and that's all we got will see you in class tomorrow

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MAT-152 7.1b: Applications of the Substitution Method

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